I had an S-Corp in IL and moved to TX. When I tried to change my address with the IL Sec of St, they told me I needed to dissolve the corp because I could not have an out-of-state mailing address. So, I dissolved the corp in IL.
My question is, can I re-use my FEIN and form an S-Corp in TX? Or do I need to start all over?

I suggest that you restart the whole process of incorporation again as your IL-corporation has been dissolved (per comment). Once the corporation has been dissolved, it is best to apply for a new EIN# when you establish another corporation.

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